Short Division: Definition, Method and Examples

What is short division?

Short division is a formal algorithm that breaks down a division problem into smaller, manageable steps. It is often called the compact division method. In some regions, it is informally known as the bus stop method because the division bracket resembles a bus shelter.

Instead of writing out long subtraction steps, short division saves space by recording the remainder from each step compactly. This method is most efficient when the divisor is a single digit or a familiar number like $12$.

Set up the division

To set up short division, arrange the numbers using a division bracket.

• Dividend: The total amount being divided goes inside the bracket.
• Divisor: The number you are dividing by goes on the outside, to the left.
• Quotient: The answer is written on top of the bracket, directly above the corresponding digits of the dividend.

Proper alignment is critical. The digits of the quotient must align perfectly with the place value columns of the dividend.

Place-value meaning of carried remainders

Short division relies heavily on place value. When you divide the digits in a column and have an amount left over, that remainder is carried to the next smaller place value.

A remainder in one column represents a bundle of ten in the next column to the right.

For example, when dividing $73$ by $3$, you first divide the $7$ tens. $7$ tens divided by $3$ gives $2$ tens, with $1$ ten left over. This $1$ ten is carried to the ones column, where it becomes $10$ ones. Combined with the existing $3$ ones, you now divide $13$ ones by $3$.

Step-by-step short division

Unlike addition or multiplication, division starts from the largest place value on the left and moves to the right.

Let's divide $536$ by $4$.

1. Divide the hundreds: $5$ hundreds divided by $4$ is $1$ hundred, with $1$ hundred remaining. Write $1$ above the hundreds place. Carry the $1$ to the tens place to make $13$ tens.
2. Divide the tens: $13$ tens divided by $4$ is $3$ tens, with $1$ ten remaining. Write $3$ above the tens place. Carry the $1$ to the ones place to make $16$ ones.
3. Divide the ones: $16$ ones divided by $4$ is exactly $4$. Write $4$ above the ones place.

The quotient is $134$.

Short division with a remainder

When a number does not divide perfectly, the final step produces a remainder. You can express division with remainders in three ways.

• Standard remainder: Write an $r$ followed by the leftover number. For example, $14 \div 4 = 3 \text{ r } 2$.
• Fraction: Use the remainder as the numerator and the divisor as the denominator. For example, $14 \div 4 = 3 \dfrac{2}{4}$.
• Decimal: Place a decimal point in the quotient and dividend, add a zero, and continue dividing. The carried remainder moves to the tenths place. For example, $14 \div 4 = 3.5$.

Short versus long division

Both methods use the exact same mathematical logic, but short division requires you to perform the subtraction step mentally.

In long division, you write out the multiplication and subtraction for each step down the page. In short division, you only write the final remainder of each step as a small digit next to the following place value.

Worked examples

Example 1: Exact division

Question: A bakery produces $855$ muffins and packs them evenly into boxes of $5$. How many boxes are needed?

Method:

1. Divide $8$ hundreds by $5$: $1$ hundred, remainder $3$. Carry the $3$.
2. Divide $35$ tens by $5$: exactly $7$ tens, remainder $0$.
3. Divide $5$ ones by $5$: exactly $1$ one.

Answer: $171$ boxes.

Check: Multiply $171$ by $5$. $100 \times 5 = 500$, $70 \times 5 = 350$, $1 \times 5 = 5$. Total is $855$. This uses the division algorithm and checking method.

Example 2: Division with a zero placeholder

Question: Calculate $618 \div 6$.

Method:

1. Divide the hundreds: $6 \div 6 = 1$. Write $1$ above the hundreds. No remainder.
2. Divide the tens: $1 \div 6 = 0$. Since $1$ is smaller than $6$, write $0$ above the tens. The $1$ ten is carried over to the ones column.
3. Divide the ones: The carried $1$ ten joins the $8$ ones to make $18$. $18 \div 6 = 3$. Write $3$ above the ones.

Answer: $103$.

Check: $103 \times 6 = 618$. The answer is correct.

Example 3: Division with a decimal remainder

Question: Share $19$ dollars equally among $4$ friends. What is the cost per person?

Method:

1. Divide $19$ by $4$: $4$ goes into $19$ four times ($16$), leaving a remainder of $3$.
2. Add a decimal point and a zero to $19$ to make $19.0$. Carry the $3$ to the tenths place to make $30$ tenths.
3. Divide $30$ by $4$: $7$ times ($28$), leaving a remainder of $2$. Carry the $2$ to the hundredths place to make $20$ hundredths.
4. Divide $20$ by $4$: exactly $5$.

Answer: Each friend pays $4.75$ dollars.

Check: $4.75 \times 4 = 19.00$ dollars.

Common mistakes

• Forgetting zero placeholders: If a digit cannot be divided by the divisor, you must place a zero in the quotient directly above it. Skipping the zero will shift all following digits to the wrong place value.
• Misaligning digits: Writing the quotient digits loosely can cause them to align with the wrong columns, leading to confusion during checking or further calculation.
• Ignoring the decimal point: When extending a division to find a decimal answer, the decimal point in the quotient must align exactly above the decimal point in the dividend.

Frequently asked questions

Can you use short division with a two-digit divisor?

Yes. You can use short division for any divisor as long as you are comfortable calculating its multiples mentally. It is very common to use short division for divisors up to $12$. For larger two-digit divisors, long division is usually less prone to mental errors.

Why do we divide from left to right?

We start with the largest place value to ensure that any remainder is broken down into smaller units (carried over) and shared among the remaining digits.